"""
Big O for various types (power series, p-adics, etc.)
"""
import sage.rings.arith as arith
import laurent_series_ring_element
import sage.rings.padics.factory as padics_factory
import sage.rings.padics.padic_generic_element as padic_generic_element
import power_series_ring_element
import integer
import rational
from sage.rings.polynomial.polynomial_element import Polynomial
import multi_power_series_ring_element
def O(*x):
"""
Big O constructor for various types.
EXAMPLES:
This is useful for writing power series elements.
sage: R.<t> = ZZ[['t']]
sage: (1+t)^10 + O(t^5)
1 + 10*t + 45*t^2 + 120*t^3 + 210*t^4 + O(t^5)
A power series ring is created implicitly if a polynomial element is passed in.
sage: R.<x> = QQ['x']
sage: O(x^100)
O(x^100)
sage: 1/(1+x+O(x^5))
1 - x + x^2 - x^3 + x^4 + O(x^5)
sage: R.<u,v> = QQ[[]]
sage: 1 + u + v^2 + O(u, v)^5
1 + u + v^2 + O(u, v)^5
This is also useful to create p-adic numbers.
sage: O(7^6)
O(7^6)
sage: 1/3 + O(7^6)
5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + 4*7^5 + O(7^6)
It behaves well with respect to adding negative powers of p:
sage: a = O(11^-32); a
O(11^-32)
sage: a.parent()
11-adic Field with capped relative precision 20
There are problems if you add a rational with very negative valuation to a big_oh.
sage: 11^-12 + O(11^15)
11^-12 + O(11^8)
The reason that this fails is that the O function doesn't know the right precision cap to use. If you cast explicitly or use other means of element creation, you can get around this issue.
sage: K = Qp(11, 30)
sage: K(11^-12) + O(11^15)
11^-12 + O(11^15)
sage: 11^-12 + K(O(11^15))
11^-12 + O(11^15)
sage: K(11^-12, absprec = 15)
11^-12 + O(11^15)
sage: K(11^-12, 15)
11^-12 + O(11^15)
"""
if len(x) > 1:
if isinstance(x[0], multi_power_series_ring_element.MPowerSeries):
return multi_power_series_ring_element.MO(x)
x = x[0]
if isinstance(x, power_series_ring_element.PowerSeries):
return x.parent()(0, x.degree())
elif isinstance(x, Polynomial):
if x.parent().ngens() != 1:
raise NotImplementedError, "completion only currently defined for univariate polynomials"
if not x.is_monomial():
raise NotImplementedError, "completion only currently defined for the maximal ideal (x)"
return x.parent().completion(x.parent().gen())(0, x.degree())
elif isinstance(x, laurent_series_ring_element.LaurentSeries):
return laurent_series_ring_element.LaurentSeries(x.parent(), 0).add_bigoh(x.valuation())
elif isinstance(x, (int,long,integer.Integer,rational.Rational)):
if x <= 0:
raise ArithmeticError, "x must be a prime power >= 2"
F = arith.factor(x)
if len(F) != 1:
raise ArithmeticError, "x must be prime power"
p, r = F[0]
if r >= 0:
return padics_factory.Zp(p, prec = max(r, 20), type = 'capped-rel')(0, absprec = r)
else:
return padics_factory.Qp(p, prec = max(r, 20), type = 'capped-rel')(0, absprec = r)
elif isinstance(x, padic_generic_element.pAdicGenericElement):
return x.parent()(0, absprec = x.valuation())
raise ArithmeticError, "O(x) not defined"
